Long-range interactions of the Sr + ion

Transcription

1 Long-range interactions of the Sr + ion J. Mitroy and J. Y. Zhang Faculty of Technology, Charles Darwin University, Darwin NT 0909, Australia M. W. J. Bromley Department of Physics and Computational Science Research Center, San Diego State University, San Diego, California 92182, USA Received 5 January 2008; published 27 March 2008 A number of properties of the low-lying states of the Sr + ion are evaluated by diagonalizing a semiempirical Hamiltonian in a large dimension single electron basis. These properties include the oscillator strengths connecting the low-lying states, the quadrupole moments, and the polarizabilities. In addition, the long-range dispersion coefficients for the states of this ion interacting with a number of atoms are given. The polarizabilities and dispersion coefficients can be used to estimate frequency shifts of the Sr + 5s 4d clock transition due to background fields, including the blackbody radiation shift, and shifts due to collisions with a buffer gas. DOI: /PhysRevA PACS number s : f, x, Nk, Bm I. INTRODUCTION This paper gives a systematic presentation of many properties of the Sr + ion, largely motivated by the importance of this ion in a number of application areas, mainly in astrophysics and cold ion physics. First, the abundance of the Sr-Y-Zr sequence of elements can reveal information about the post-main-sequence evolution of cool giant stars. The lifetimes and other data of Sr + are useful in determining strontium elemental abundances 2,3. Second, the long lifetime of the 5s 4d transition of the 88 Sr + ion makes it a promising candidate for a trapped ion frequency standard 4 8. Various atomic properties are useful in evaluating the uncertainty budget, for example, blackbody radiation shift is predominantly determined by the difference between the polarizabilities of the 4d and 5s levels. In addition, there have been a number of investigations of pressure broadening of the Sr + transitions in a variety of buffer gases 2,9 11. Knowledge of the long-range part of the ion-atom interaction is useful in determining the pressure broadening and shift 12,13. Finally, the Sr + Ar and Sr + Kr molecules have been synthesized and the present information can improve the characterizations of the potential surface 14,15. In the present work, a one-electron model of the Sr + ion is constructed with a semiempirical core potential to describe the interaction of the valence electron with the rest of the atom. The polarizabilities and dispersion coefficients are evaluated using sum rules 16,17. All quantities are reported in atomic units, with the exception of lifetimes and decay rates which are stated in SI units. II. THEORETICAL DEVELOPMENT A. Overview of dispersion constant calculations The long-range interaction between one electrically charged atom and one electrically neutral atom has two components when terms up to second order in perturbation theory are included. There is the polarization interaction, which can be written V R = d 2R 4 q 2R 6, where d and q are the dipole and quadrupole polarizabilities of the neutral atom. The distance between the two nuclei is R. This part of the long-range interaction does not lead to a frequency shift between the different states of the ion. There is also the dispersion interaction. For two spherically symmetric atoms in their ground states, this interaction can be written 21,22 as V R = C 6 R 6 C 8 R 8 C 10 R 10. The C n parameters are the dispersion coefficients. The dispersion interaction leads to a frequency shift between the different ion states when the ion is immersed in a buffer gas. The approach used to generate the dispersion coefficients 16,17 is based on the work of Dalgarno who derived expressions in terms of oscillator strength sum rules 21,22. This reduced the calculation of the C n parameters for two spherically symmetric atoms to sums over the products of the absorption oscillator strengths originating in the ground state divided by an energy denominator. The sums should include contributions from all discrete and continuum excitations. In practice a pseudostate representation is used which gives a discrete representation of the continuum 17,23,24. Finite dimension sums over a pseudostate basis give a rapidly convergent expansion of the continuum of intermediate states provided all the pseudostates are retained The sum over the oscillator strengths needs to be rewritten in terms of a sum over the reduced matrix elements of the electric multipole operator in cases where one or both of the atoms is in a state with L The major part of any calculation involves the generation of the lists of reduced transition matrix elements for the two atomic states. In the present approach this is done by diagonalizing a semiempirical model potential in a large single electron basis. This gives a discretization of the positive energy continuum, as well as the spectrum of low-lying physi /2008/77 3 / The American Physical Society

2 MITROY, ZHANG, AND BROMLEY cal states. It is then straightforward to process the lists of matrix elements and generate the dispersion coefficients 17. B. Wave functions and transition operators for Sr + The wave functions and transition operators were computed by diagonalizing the semiempirical Hamiltonian 24,28 31 in a large mixed Laguerre-type orbital LTO and Slater-type orbital STO basis set 24. The initial step was to perform a Hartree-Fock HF calculation to define the core. In this case, calculations of the Sr 5s 2 ground states were done in a STO basis 32. The core wave functions were then frozen, giving the working Hamiltonian for the valence electron, H = V dir r + V exc r + V p r. The direct and exchange interactions of the valence electron with the HF core were calculated exactly. The -dependent polarization potential, V p, was semiempirical in nature with the functional form V p r = d g 2 r m 2r 4 m m. 4 The coefficient, d is the static dipole polarizability of the core and g 2 r =1 exp r 6 / 6 is a cutoff function designed to make the polarization potential finite at the origin. The cutoff parameters, were tuned to reproduce the binding energies of the ns ground state and the np, nd, and nf excited states. The dipole polarizability for Sr 2+ was chosen as d =5.813a ,33. The cutoff parameters for =0 3 were 1.746, , , and a 0, respectively. The energies of the states with 1 were assigned to the statistical average of their respective spin-orbit doublets. The Hamiltonian was diagonalized in a very large orbital basis with about 50 Laguerre-type orbitals for each value. Although the wave functions are constructed as linear combinations of LTOs, all matrix element evaluations were done using Gaussian quadratures and are accurate to close to machine precision. The cutoff parameters were different for each 3. The parameters for 3 were set to 3. The oscillator strengths and other multipole expectation values were computed with operators that included polarization corrections 24,29, The quadrupole core polarizability was chosen as 17.15a ,33 while the octupole polarizability was set to zero. The cutoff parameter for the polarization corrections to transition operator was fixed at 2.16a 0 the average of 0, 1, and 2. The model potential is realistic since the direct and exchange interactions with the core were computed without approximation from the HF wave function, and only the core polarization potential is described with a model potential. Comparisons with more sophisticated ab initio methods have revealed that the present semiempirical approach often gives oscillator strengths, polarizabilities, and dispersion coefficients that lie within a couple of percent of the best calculations 24, There is one complication. The 4d orbital 3 of Sr + is compact and does penetrate into the core like the 3d orbital of Ca + 40,41. Defining the polarization potential by tuning to the 4d energy leads to the rest of the nd series being slightly under bound, and then there is the issue of the accuracy of the 4d state itself. Consequently, quantities which explicitly involve the Sr + nd set of states can be expected to have larger uncertainties than the other states. C. Wave functions and transition operators or other atoms The transition arrays for hydrogen and the rare gases are exactly the same as those in previous investigations of the dispersion interactions involving the low-lying states of the alkali-metal atoms and magnesium with these atoms 38,42. The arrays for hydrogen and helium should be regarded as being capable of giving essentially exact polarizabilities and dispersion coefficients. III. RESULTS A. Energy levels The binding energies of the low-lying states of the Sr + are tabulated and compared with experiment in Table I. The agreement between the present energies and the experimental energies is generally of order 10 4 hartree with the exception of the 5d level where there is a discrepancy of hartree. The value of 2 was tuned to the 4d energy which penetrates into the core 28,36,40,41. Tuning the cutoff parameter to the 4d level leads to the binding energy of the 5d and higher levels being underestimated 28,40. B. Oscillator strengths The absorption oscillator strength for a multipole transition from n a n b, with an energy difference of E nb n a =E nb E na, is defined as k f na n b TABLE I. Theoretical and experimental energy levels in hartree of some of the low-lying states of the Sr + ion. The energies are given relative to the energy of the Sr 2+ core. The experimental energies taken from 43 for the doublet states are averages with the usual 2J+1 weighting factors. State Theory Experiment 5s d p s d p f = 2 n a ;L a r k C k rˆ nb ;L b 2 E nb n a. 5 2k +1 2L a +1 In this expression, L a is the orbital angular momentum of the initial state while k is the polarity of the transition. Table II

3 LONG-RANGE INTERACTIONS OF THE Sr + ION TABLE II. Absorption oscillator strengths for various transitions of Sr +. All of the oscillator strengths are for dipole transitions with three exceptions. Theoretical oscillator strengths from other groups are identified by an acronym denoting the type of calculation and a citation. The numbers in the square brackets in the column labeled Present denote powers of 10. Transition Present Other f 5s 5p f 5s 6p f 4d 5p f 4d 6p f 4d 4f f 5p 6s f 5p 5d f 5d 4f 2 f 5s 4d 2 f 5s 5d CA, Ref. 44 ; MCHF, Ref. ; MBPT, Ref CA, Ref CA, Ref. 44 ; MCHF, Ref. ; MBPT, Ref CA, Ref CA, Ref. 44 ; MCHF, Ref CA, Ref CA, Ref CA, Ref lists the present oscillator strengths for a number of Sr + transitions. Values from three other calculations are listed 1,44,45 for the dipole transitions. The Coulomb approximation CA values are mainly included because they are a part of a set that covers all the transitions. More sophisticated calculations are the multiconfiguration Hartree-Fock MCHF calculation of Brage et al. and the many-body perturbation theory MBPT calculation by Guet and Johnston 45. There is a 3% variation among the calculations of the f 5s 5p oscillator strength. The MCHF and MBPT calculations were both relativistic and the values given in Table II are the sums of the f values of the spin-orbit doublet. Although the lifetime for the 5p level has been measured on numerous occasions 46,47, it is not possible to directly determine the 5s 5p oscillator strength from this information since the 5p level can decay into both the 4d and 5s levels. Accordingly, we have converted the present oscillator strengths listed in Table II into a 5p lifetime of s for comparison with experiment. The most recent beam-foil experiment gave s and s 47 for the lifetime of the 5p 1/2 and 5p 3/2 states, respectively. Previously, another beam-foil experiment gave s and s 48 for the lifetimes of the 5p 1/2 and 5p 3/2 states. Taking a statistically weighted average of the experimental values would result in average lifetimes that are within a couple of percent of the present lifetime. This level of accuracy is the best that can be expected from an underlying theoretical model that does not allow for spin-orbit splitting. The branching ratio for the decay to the 5s and 4d levels have also been determined 46. For the decay from the 5p 3/2 level the branching ratio was while the ratio for the 5p 1/2 state was The present calculation gives a branching ratio of Other calculations give branching ratios that range between 16.0 to ,45. A more precise experimental determination of this branching ratio would allow better discrimination between the different models of the Sr + structure. There is a surprisingly large difference between the MCHF and present value of f 4d 4f with the present calculation yielding and the MCHF calculation yielding This problem can be resolved by reference to the experimental lifetimes from a beam foil experiment 47. The lifetime of the 4f level has been measured as ns and ns for the 4f 5/2 and 4f 7/2 states, respectively 47. The current calculations give a 4f lifetime of 3.00 ns. Although the 4f level can also decay to the 5d level, the total decay rate is dominated by the 4d 4f transition and the 4d:5d branching ratio is The experimental 4f lifetime thus suggests that the present f 4d 4f should be preferred to the MCHF value. C. Lifetimes of the 4d levels The transition rate for the 5s 4d transition has been the subject of many recent investigations 49 52, and the oscillator strength for this transition is also given in Table II. The decay rate of an LS coupled state decaying by a quadrupole transition is related to the reduced matrix element by the identity 53 A E2 = L b +1 5 n a ;L a r k C k rˆ nb ;L b 2, where is given in Å. The present calculation using the theoretical energy difference gives a lifetime of s. The lifetime is sensitive to small changes in the transition energy as it depends on the fifth power of the transition energy. If the lifetime is computed with the actual energy differences of the spin-orbit doublet, we obtain 3/2 =0.443 s and 5/2 =0.404 s. It should be noted that magnetic dipole decays from the 4d 3/2 and 4d 5/2 states are possible, but they have small decay rates and a very small effect on the lifetimes 45,49. There is some scatter between even some recent estimates of the 4d 3/2 lifetimes. The three recent lifetime measurements gave s 50, s 54, and s 54 while an older experiment gave s 55. The relativistic MBPT calculation of Guet and Johnson gave 3/2 =0.454 s 1 45, while a relativistic coupled cluster method with single and double excitations CCSD gave 3/2 =0.426 s 49. The nature of the scatter among the different estimates of 5/2 is bimodal in nature. There is one set of estimates that predict a lifetime of about 0.40 s, while another set of estimates predict a lifetime of about 0.36 s. For example, the most recent experiments gave 5/2 = s 54 and 5/2 = s 52. However some older experiments have given 5/2 = s 55, 5/2 = s 56, and 5/2 = s 57. The relativistic MBPT calculation gave 5/2 =0.405 s 45 while the CCSD calculation gave 5/2 =0.357 s

4 MITROY, ZHANG, AND BROMLEY TABLE III. The dipole, quadrupole, and tensor polarizabilities for dipole excitations for the low-lying states of Sr +. A description of the other calculations can be found in the text of Ref. 2. The quadrupole moments are computed for L=M which are the same as the J=M J =L+1/2 state. The numbers in the square brackets denote powers of a.u. 1,2LL a.u. 2 a.u. 3 a.u. Q LL State Present Other Present Present Present Present 5s f sums, Ref d f sums, Ref p f sums, Ref s d p There is a significant difference in the results of the CCSD 49 calculations and the MBPT calculations 45. The ratio of the two lifetimes for the MBPT calculation is while the ratio of the two lifetimes for the CCSD calculation is Assuming that the reduced matrix elements connecting the 4d and 5s states are the same when angular momentum factors are taken into account, Eq. 6 suggests the lifetimes should scale as the fifth power of the energy difference and this gives a ratio of Examining Table 1 of Sahoo et al. 49 reveals that the lifetime ratio of the 3d levels for Ca + is while a value of is expected from energy factors alone. Two relativistic MBPT calculations gave and Further, the 2 parameter in the present calculation was changed to give the 4d state a binding energy equal to the 4d 5/2 experimental energy. The resulting change in the radial distribution of the 4d orbital caused the 4d 5/2 lifetime to decrease by only 0.37%. The results of the present and MBPT calculations suggest that the CCSD calculations of Sahoo et al. are tending to systematically overestimate the differences between the nd 5/2 and nd 3/2 lifetimes for Ca + and Sr +. A similar conclusion was previously reached by Kreuter et al. 58 after an extensive analysis. D. Quadrupole moments Of the quadrupole moments given in Table III, the one of most interest is that of the 4d state. This is due to the possible use of the Sr + 4d 5s transition as an optical frequency standard 8,59. The quadrupole moment of this state has been determined in two recent large-scale calculations 59,60. The relativistic configuration interaction RCI calculations of Itano et al. 59 gave quadrupole moments of Q 3/2 =2.11 a.u. and Q 5/2 =3.02 a.u. for the 4d 3/2 and 4d 5/2 states, respectively. The relativistic coupled-cluster RCC approach gave quadrupole moments of Q 3/2 =2.12 a.u. and Q 5/2 =2.94 a.u. 60. A recent experiment has determined the quadrupole moment, Q 5/2,tobe2.6 3 a.u. 61. The present estimate of the quadrupole moment for the 4d level is a.u., which is 3 5 % smaller than the two calculations, but closer to the recent experiment. This is reminiscent of the situation for Ca + where a semiempirical model potential method 16 gave a quadrupole moment smaller than the two ab initio calculations 59,60 and in better agreement with experiment. Examination of the convergence pattern of the RCI and RCC calculation reveals that substantial cancellations exist between some quite large terms and relatively small imperfections of the calculation of any of these terms could translate to errors in the final quadrupole moment. The present calculation may not be definitive, but there is less scope for error within the framework of the frozen core approximation and it should be preferred to the RCC and RCI estimates of the quadrupole moments. E. Polarizabilities The polarizabilities of the lowest six levels of Sr + are listed in Table III and compared with the results of other calculations. The dipole polarizability for the ground state, 89.88a 3 0 is about 3% smaller than the estimate of 93.3a 3 0 derived from oscillator strength sums using experimental data 2 note, the core polarizability of 5.8a 3 0 was added to the values listed in 2. The uncertainty in the present polarizability for the ground state should be about 2 3 % and this mainly results from the uncertainty in the 5s 5p oscillator strength. The polarizability for the 4d state is smaller than that of the ground state because there are no strong dipole transitions from the 4d state to the other low-lying levels. The negative polarizability for the 5p state is caused by the transitions to the states of lower energy. The respective size of the polarizabilities of the 5s, 4d, and 5p levels are similar to the respective sizes of the polarizabilities for the 4s, 3d, and 4p levels for Ca F. Blackbody shift The difference between the polarizabilities of the 5s and 4d states contributes to the blackbody shift through the approximate result 62 = 1 4 T K V/m 2 d 4d d 5s The polarizabilities in Eq. 7 are assumed to be in SI units which can be obtained by multiplying the values in Table III

5 LONG-RANGE INTERACTIONS OF THE Sr + ION TABLE IV. The dispersion coefficients in a.u. between the Sr + 5s and Sr + 6s states and the rare-gas atoms and atomic hydrogen. The Barklem et al. estimates 2 used oscillator strength sums to evaluate the dispersion constants. The numbers in the square brackets denote powers of 10. System C 6 C 8 C 10 Sr + 5s -H f sums, Ref Sr + 5s -He Sr + 5s -Ne Sr + 5s -Ar Sr + 5s -Kr Sr + 5s -Xe Sr + 6s -H Sr + 6s -He Sr + 6s -Ne Sr + 6s -Ar Sr + 6s -Kr Sr + 6s -Xe by Using the present polarizabilities gives =0.242 Hz at 300 K. One recent estimate of the frequency shift by Madej et al. gave Hz 8. Their estimate of the frequency shift is larger than the present value because Madej et al. underestimated the polarizability of the 4d states since they did not include the contribution from the continuum. Their value of 54a 3 0 a core polarizability of 5.8a 3 0 was added to the number reported in 8 is about 10% smaller than the present polarizability of 61.77a 3 0. The possible influence of the spin-orbit splitting on the blackbody shift is estimated by replacing the spin-orbit averaged energy of the 2 D doublet with the actual energy with the 2 D 5/2 level at hartree 43. When this was done, the 4d polarizability increased to 62.92a 3 0 since the energy difference between the 4d 5/2 and the 5p levels is slightly smaller. The net effect of the slightly largely polarizability is a 4% reduction in the blackbody shift to Hz at 300 K. G. Dispersion coefficients with buffer atoms The dispersion coefficients of a number of low-lying states interacting with hydrogen and the rare gases are listed in Tables IV VI. The dispersion coefficients involving the Sr + ground state are listed in Table IV. Dispersion coefficients involving the 4d and 5p excited states are given in Tables V and VI, respectively. There is relatively little data available for comparison 2. The C 6 estimate of Barklem et al. 2 for the Sr + 5s -H configuration uses the Sr + oscillator strengths which are taken from other sources. As such, there is no contribution from the Sr 2+ core. The hydrogen oscillator strength distribution is based on an Unsold-type approximation 2,63 where the entire distribution is represented by a single term TABLE V. The dispersion coefficients in a.u. between the 4d states of Sr + and the rare-gas atoms and atomic hydrogen. The Barklem et al. estimates 2 used oscillator strength sums to evaluate the dispersion constants and is assumed to represent an average over possible magnetic quantum numbers. The numbers in the square brackets denote powers of 10. System C 6 C 8 C 10 Sr + 4d -H Sr + 4d -H Sr + 4d -H f sums, Ref Sr + 4d -He Sr + 4d -He Sr + 4d -He Sr + 4d -Ne Sr + 4d -He Sr + 4d -He Sr + 4d -Ar Sr + 4d -Ar Sr + 4d -Ar Sr + 4d -Kr Sr + 4d -Kr Sr + 4d -Kr Sr + 4d -Xe Sr + 4d -Xe Sr + 4d -Xe with an appropriately chosen excitation energy. The present estimate of C 6 =62.70 a.u. is about 20% larger than the Barklem et al. estimate of C 6 =51.9 a.u. The core contribution, which was omitted by Barklem et al., was about 20%. The dispersion coefficients for the 4d states are larger than the value of Barklem et al. 2 which can be interpreted as an average over the different molecular symmetries. This is due to the omission of the 4d continuum in the intermediate sum of Barklem et al. 2. Barklem et al. postulate a maximum value of 32.8 a.u. can arise from the subsequent inclusion of the valence state continuum. The dispersion coefficients of the 4d level are generally smaller than the dispersion coefficients of the 5s level. This is similar to the situation in Ca + where the dispersion coefficients for the 3d level were generally smaller than those for the 4s ground state. This occurs because there are no strong transitions between the 4d level and the other low-lying levels of Sr + refer to dipole oscillator strengths in Table II. The lack of any strong transitions to the low-lying states also manifested itself in the smaller polarizabilities of the 4d state. Although the dipole polarizability of the 5p state is smaller than the polarizability of the 5s state, the dispersion coefficients involving the 5p state are larger than those involving the ground state. This is explained by the energy distribution of the 5p oscillator strength set. The transitions to the 4d and 5s states with negative oscillator strengths involve smaller transition energies than any of the upward

6 MITROY, ZHANG, AND BROMLEY TABLE VI. The dispersion coefficients in a.u. between the Sr + 5p state and the rare-gas atoms and atomic hydrogen. The Barklem et al. estimates 2 used oscillator strength sums to evaluate the dispersion constants and is assumed to represent an average over possible magnetic quantum numbers. The numbers in the square brackets denote powers of 10. System C 6 C 8 C 10 Sr + 5p -H Sr + 5p -H f sums, Ref Sr + 5p -He Sr + 5p -He Sr + 5p -Ne Sr + 5p -Ne Sr + 5p -Ar Sr + 5p -Ar Sr + 5p -Kr Sr + 5p -Kr Sr + 5p -Xe Sr + 5p -Xe transitions. So the impact of the downward transitions is maximized in the polarizability calculation with its 1/ E 2 energy denominator. However, the downward transitions have less impact in the C 6 calculation which uses different energy weighting factors. IV. CONCLUSIONS A survey of atomic parameters of the Sr + ion states relevant to the description of long-range interactions have been computed with a semiempirical method. The derived parameters have application in a number of different areas of physics. The main limitation of the present method is the absence of spin-orbit splitting and the results given here should be regarded as averages. One aspect of the present calculation that may be regarded as unusual is that in a number of instances the present results are in better agreement with experiment than the results of apparently more sophisticated ab initio calculations. Two specific examples are the 4d 4f oscillator strength and the 4d quadrupole moment. It must be kept in mind that the polarization potentials are tuned to the experimental energies. Therefore, the radial dependence of the single electron wave functions are guaranteed to be correct at large distances. Many of the matrix elements relevant to the observables reported here are strongly weighted by the large-r form of the wave function. So tuning the model energy levels to experiment does go a long way to ensuring many other of the interesting observables will be predicted accurately. It would clearly be interesting to apply the present semiempirical approach within a relativistic framework. ACKNOWLEDGMENTS The authors would like to thank Shane Caple and Corey Hoffman of CDU for workstation support. T. Brage, G. M. Wahlgren, S. G. Johansson, D. S. Leckrone, and C. R. Proffitt, Astrophys. J. 496, P. S. Barklem and B. J. O Mara, Mon. Not. R. Astron. Soc. 311, L. Mashonkina and T. Gehren, Astron. Astrophys. 376, G. P. Barwood, G. Huang, H. A. Klein, P. Gill, and R. B. M. Clarke, Phys. Rev. A 59, R H. S. Margolis, G. Huang, G. P. Barwood, S. N. Lea, H. A. Klein, W. R. C. Rowley, P. Gill, and R. S. Windeler, Phys. Rev. A 67, P. Gill, G. P. Barwood, H. A. Klein, G. Huang, S. A. Webster, P. J. Blythe, K. Hosaka, S. N. Lea, and H. S. Margolis, Meas. Sci. Technol. 14, H. S. Margolis, G. P. Barwood, G. Huang, H. A. Klein, S. N. Lea, K. Szymaniec, and P. Gill, Science 306, A. A. Madej, J. E. Bernard, P. Dubé, L. Marmet, and R. S. Windeler, Phys. Rev. A 70, R. G. Giles and E. L. Lewis, J. Phys. B 15, H. Harima, T. Ihara, Y. Urano, and K. Tachibana, Phys. Rev. A 34, H. Harima, K. Tachibana, and Y. Urano, Phys. Rev. A 35, G. Peach, Adv. Phys. 30, N. Allard and J. Kielkopf, Rev. Mod. Phys. 54, S. S. Xantheas, G. S. Fanourgakis, S. C. Farantos, and M. Velegrakis, J. Chem. Phys. 108, D. Prekas, B.-H. Feng, and M. Velegrakis, J. Chem. Phys. 108, J. Mitroy and J. Y. Zhang, Eur. Phys. J. D 46, J. Y. Zhang and J. Mitroy, Phys. Rev. A 76, H. Kreek and W. J. Meath, J. Chem. Phys. 50, R. Ahlrichs, H. J. Bohm, S. Brode, K. T. Tang, and J. P. Toennies, J. Chem. Phys. 88, R. Côté and A. Dalgarno, Phys. Rev. A 62, A. Dalgarno and W. D. Davison, Adv. At. Mol. Phys. 2, A. Dalgarno, Adv. Chem. Phys. 12, J. F. Bukta and W. J. Meath, Mol. Phys. 27, J. Mitroy and M. W. J. Bromley, Phys. Rev. A 68, C. K. Au and R. J. Drachman, Phys. Rev. A 37, A. K. Bhatia and R. J. Drachman, Phys. Rev. A 45, R. J. Drachman and A. K. Bhatia, Phys. Rev. A 51, J. Mitroy, D. C. Griffin, D. W. Norcross, and M. S. Pindzola, Phys. Rev. A 38, J. Mitroy, J. Phys. B 26, M. W. J. Bromley and J. Mitroy, Phys. Rev. A 65,

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